320 OBSERVATIONS on THE THEORY 
The maximum effect of an underfhot wheel, produced 
by a given quantity of water in a given time, is in the 
duplicate ratio of the velocity of the water. For the effect 
mutt be as the impetus acting on the wheel, multiplied in- 
to the velocity of the wheel. But this zmpetus 1s demon- 
{trated to be, fimply, as the relative velocity, [Prop. 1. 
page 146] and the velocity of the wheel producing a max- 
imum, being half that of the water, [by Prop. II. page 
14.7.] is likewife as the velocity of the water: therefore, 
the power ating on the wheel multiplied into the wheel’s 
velocity, or the effe@ produced, muft be in the duplicate 
ratio of the velocity of the water. Q. E. D. 
Corol. Hence the effe& produced by a given quantity 
of water in a given time, will be as the height of the 
head, becaufe this height is as the {quare of the velocity. 
This alfo agrees with experiment. 
If the force acting on the wheel, were in the duplicate 
ratio of the water’s velocity, as ufually afferted, then the 
effect would be as the cube thereof, when the quantity of 
water and time are given; which is contrary to the refult 
of experiment. 
When I attempted to compute the power, &c. of Door 
Barker’s or James Rumfey’s mill, as at page 185, the 
following fimple demonftration of its equivalence to that 
of the underfhot wheel, with the fame quantity and fall of 
water, had not then occurred, viz. 
Adiion and re-adtion are equal, &c. 
But the underfhot wheel is propelled by the action, and 
the rotatory tube by the re-acfion of the fame agent or mo- 
mentum. 
Therefore their mechanical effects muft be equal. 
The ating and counteracting forces which originate 
infide the tube, not being from any external impulfe, can 
be no exception; becaufe, in any body, or fyftem of bodies, 
the 
